The structure of your data matters a lot. If one will fit in cache but the other won't, the one that fits in cache will almost always win. Then they have different big-O characteristics: binary searching an array is O(log N) where N is the number of items, a trie is O(k) where k is the length of the key, a hashtable is O(1), but this is misleading because the hash function is usually O(k) and performance can degrade to O(n) if you have lots of hash collisions.
I thought that an autocompletion widget for stock tickers would be the perfect application for tries: short, dense key space, lots of elements, and a fair likelihood of hash collisions. But apparently not, because our data size just happened to be one where cache effects dominate. I talked to one of the GMail guys later and he was quite surprised, from which I inferred (but it was not stated) that GMail probably uses tries to good effect.